Page 409 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

THEOREM 5.5.4
Elementary row operations do not change the row space of a matrix.

Proof Suppose that the row vectors of a matrix A are , , … , , and let B be obtained from A by performing an elementary
row operation. We shall show that every vector in the row space of B is also in the row space of A and that, conversely, every
vector in the row space of A is in the row space of B. We can then conclude that A and B have the same row space.

Consider the possibilities: If the row operation is a row interchange, then B and A have the same row vectors and consequently
have the same row space. If the row operation is multiplication of a row by a nonzero scalar or the addition of a multiple of one
row to another, then the row vectors , , …, of B are linear combinations of , , …, ; thus they lie in the row space of A.
Since a vector space is closed under addition and scalar multiplication, all linear combinations of , , …, will also lie in the
row space of A. Therefore, each vector in the row space of B is in the row space of A.

Since B is obtained from A by performing a row operation, A can be obtained from B by performing the inverse operation (Section
1.5). Thus the argument above shows that the row space of A is contained in the row space of B.

In light of Theorems Theorem 5.5.3 and Theorem 5.5.4, one might anticipate that elementary row operations should not change the
column space of a matrix. However, this is not so— elementary row operations can change the column space. For example,
consider the matrix

The second column is a scalar multiple of the first, so the column space of A consists of all scalar multiples of the first column
vector. However, if we add −2 times the first row of A to the second row, we obtain

Here again the second column is a scalar multiple of the first, so the column space of B consists of all scalar multiples of the first
column vector. This is not the same as the column space of A.

Although elementary row operations can change the column space of a matrix, we shall show that whatever relationships of linear

independence or linear dependence exist among the column vectors prior to a row operation will also hold for the corresponding

columns of the matrix that results from that operation. To make this more precise, suppose a matrix B results from performing an

elementary row operation on an matrix A. By Theorem 5.5.3, the two homogeneous linear systems

have the same solution set. Thus the first system has a nontrivial solution if and only if the same is true of the second. But if the
column vectors of A and B, respectively, are

then from 2 the two systems can be rewritten as

(6)
and

(7)
Thus 6 has a nontrivial solution for , , …, if and only if the same is true of 7. This implies that the column vectors of A are
linearly independent if and only if the same is true of B. Although we shall omit the proof, this conclusion also applies to any
subset of the column vectors. Thus we have the following result.